# American Institute of Mathematical Sciences

January  2015, 2(1): 51-64. doi: 10.3934/jcd.2015.2.51

## An elementary way to rigorously estimate convergence to equilibrium and escape rates

 1 Dipartimento di Matematica Applicata, Università di Pisa, Via Bonanno Pisano 2 Instituto de Matemática, UFRJ Av. Athos da Silveira Ramos 149, Centro de Tecnologia, Bloco C Cidade Universitária, Ilha do Fundão, Caixa Postal 68530 21941-909 Rio de Janeiro, RJ, Brazil 3 Laboratoire de Mathématiques, CNRS UMR 6205, Université de Bretagne Occidentale, 6 av. Victor Le Gorgeu, CS 93837, 29238 BREST Cedex 3

Received  April 2014 Revised  January 2015 Published  August 2015

We show an elementary method to obtain (finite time and asymptotic) computer assisted explicit upper bounds on convergence to equilibrium (decay of correlations) and escape rates for systems satisfying a Lasota Yorke inequality. The bounds are deduced from the ones of suitable approximations of the system's transfer operator. We also present some rigorous experiments on some nontrivial example.
Citation: Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51
##### References:
 [1] V. Araujo, S. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Mathematiche Zeitcrift, 276 (2014), 1001-1048. doi: 10.1007/s00209-013-1231-0. [2] W. Bahsoun, C. Bose and G. Froyland, (Eds.), Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proceedings in Mathematics & Statistics, 70. Springer, New York, 2014. doi: 10.1007/978-1-4939-0419-8. [3] W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529-2542. doi: 10.1088/0951-7715/19/11/002. [4] W. Bahsoun and C. Bose, Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty }$, Nonlinear Analysis, 74 (2011), 4481-4495. doi: 10.1016/j.na.2011.04.012. [5] V. Baladi and M. Holschneider, Approximation of nonessential spectrum of transfer operators, Nonlinearity Nonlinearity, 12 (1999), 525-538. doi: 10.1088/0951-7715/12/3/006. [6] L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463. doi: 10.1007/s002200100427. [7] C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes,, , (). [8] M. D. Boshernitzan, Quantitative recurrence results, Inv. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320. [9] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Elsevier, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1. [10] G. Froyland, Extracting dynamical behaviour via Markov models, in Alistair Mees, editor, Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, (Cambridge 1998), 281-321, Birkhauser, 2001. [11] G. Froyland, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), 237-257. doi: 10.1007/s002200050198. [12] S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures, SIAM J. Appl Dyn Sys., 13 (2014), 958-985. doi: 10.1137/130911044. [13] S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805. doi: 10.4310/MRL.2007.v14.n5.a8. [14] S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps,, , (). [15] B. Hunt, Estimating invariant measures and Lyapunov exponents, Erg. Th. Dyn. Sys., 16 (1996), 735-749. doi: 10.1017/S014338570000907X. [16] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152. [17] O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations, SIAM J. Applied Dynamical Systems, 10 (2011), 788-816. doi: 10.1137/09077864X. [18] O. E. Lanford III, Informal remarks on the orbit structure of discrete approximations to chaotic maps, Exp. Math., 7 (1998), 317-324. doi: 10.1080/10586458.1998.10504377. [19] A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1. [20] C. Liverani, Rigorous numerical investigations of the statistical properties of piecewise expanding maps-A feasibility study, Nonlinearity, 14 (2001), 463-490. doi: 10.1088/0951-7715/14/3/303. [21] C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica "Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa, 2004.

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##### References:
 [1] V. Araujo, S. Galatolo and M. J. Pacifico, Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors, Mathematiche Zeitcrift, 276 (2014), 1001-1048. doi: 10.1007/s00209-013-1231-0. [2] W. Bahsoun, C. Bose and G. Froyland, (Eds.), Ergodic Theory, Open Dynamics, and Coherent Structures, Springer Proceedings in Mathematics & Statistics, 70. Springer, New York, 2014. doi: 10.1007/978-1-4939-0419-8. [3] W. Bahsoun, Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), 2529-2542. doi: 10.1088/0951-7715/19/11/002. [4] W. Bahsoun and C. Bose, Invariant densities and escape rates: Rigorous and computable approximations in the $L^{\infty }$, Nonlinear Analysis, 74 (2011), 4481-4495. doi: 10.1016/j.na.2011.04.012. [5] V. Baladi and M. Holschneider, Approximation of nonessential spectrum of transfer operators, Nonlinearity Nonlinearity, 12 (1999), 525-538. doi: 10.1088/0951-7715/12/3/006. [6] L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys., 219 (2001), 443-463. doi: 10.1007/s002200100427. [7] C. Bose, G. Froyland, C. Gonzales-Tokman and R. Murray, Ulam's Method for Lasota Yorke maps with holes,, , (). [8] M. D. Boshernitzan, Quantitative recurrence results, Inv. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320. [9] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of dynamical systems, Elsevier, 2 (2002), 221-264. doi: 10.1016/S1874-575X(02)80026-1. [10] G. Froyland, Extracting dynamical behaviour via Markov models, in Alistair Mees, editor, Nonlinear Dynamics and Statistics: Proceedings, Newton Institute, (Cambridge 1998), 281-321, Birkhauser, 2001. [11] G. Froyland, Computer-assisted bounds for the rate of decay of correlations, Comm. Math. Phys., 189 (1997), 237-257. doi: 10.1007/s002200050198. [12] S. Galatolo and I. Nisoli, An elementary approach to rigorous approximation of invariant measures, SIAM J. Appl Dyn Sys., 13 (2014), 958-985. doi: 10.1137/130911044. [13] S. Galatolo, Dimension and hitting time in rapidly mixing systems, Math. Res. Lett., 14 (2007), 797-805. doi: 10.4310/MRL.2007.v14.n5.a8. [14] S. Galatolo and I. Nisoli, Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps,, , (). [15] B. Hunt, Estimating invariant measures and Lyapunov exponents, Erg. Th. Dyn. Sys., 16 (1996), 735-749. doi: 10.1017/S014338570000907X. [16] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152. [17] O. Ippei, Computer-assisted verification method for invariant densities and rates of decay of correlations, SIAM J. Applied Dynamical Systems, 10 (2011), 788-816. doi: 10.1137/09077864X. [18] O. E. Lanford III, Informal remarks on the orbit structure of discrete approximations to chaotic maps, Exp. Math., 7 (1998), 317-324. doi: 10.1080/10586458.1998.10504377. [19] A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. doi: 10.1090/S0002-9947-1973-0335758-1. [20] C. Liverani, Rigorous numerical investigations of the statistical properties of piecewise expanding maps-A feasibility study, Nonlinearity, 14 (2001), 463-490. doi: 10.1088/0951-7715/14/3/303. [21] C. Liverani, Invariant Measures and Their Properties. A Functional Analytic Point of View, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Centro di Ricerca Matematica "Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa, 2004.
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